9-1
第 9章资本资产定价模型
The Capital Asset Pricing Model
9-2
9.1 股票的需求与均衡价格
9.2 资本资产定价模型
9.3 资本资产定价模型的扩展形式
9.4 资本资产定价模型与流动性资本资产定价模型
Capital Asset Pricing Model (CAPM)
9-3
The supply and demand for shares determine equilibrium
prices and expected rates of return,Imagine a simple world
with only two corporations,Bottom Up Inc,(BU) and Top
Down Inc,(TD),Stock prices and market values are shown
in Table 9.1,Investors can also invest in a money market
fund (MMF) which yields a risk-free interest rate of 5%.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-4
Sigma Fund is a new actively managed mutual fund that has
raised $220 million to invest in the stock market,The security
analysis staff of Sigma believes that neither BU nor TD will
grow in the future and therefore,that each firm will pay level
annual dividends for the foreseeable future,This is a useful
simplifying assumption because,if a stock is expected to pay
a stream of level dividends,the income derived from each
share is a perpetuity,Therefore,the present value of each
share often called the intrinsic value of the share equals the
dividend divided by the appropriate discount rate,A summary
of the report of the security analysts appears in Table 9.2.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-5
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-6
Using these data and assumptions Sigma easily
generates the efficient frontier shown in Figure 9.1
and computes the optimal portfolio proportions
corresponding to the tangency portfolio,These
proportions,combined with the total investment
budget,yield the fund’s buy orders,With a budget of
$220 million,Sigma wants a position in BU of
$220,000,000 X 0.8070 =$177,540,000,or
$177,540,000/39 =4,552,308 shares,and a position
in TD of $220,000,000 X 0.1930= $42,460,000,
which corresponds to 1,088,718 shares.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-7
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-8
The expected rates of return that Sigma used to
derive its demand for shares of BU and TD were
computed from the forecast of year-end stock
prices and the current prices,If,say,a share of
BU could be purchased at a lower price,Sigma’s
forecast of the rate of return on BU would be
higher,Conversely,if BU shares were selling at a
higher price,expected returns would be lower,A
new expected return would result in a different
optimal portfolio and a different demand for shares.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-9
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-10
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
Sigma’s demand curve for BU stock is given by the
Desired Shares column in Table 9.3 and is plotted in
Figure 9.2,Notice that the demand curve for the stock
slopes downward,When BU’s stock price falls,Sigma will
desire more shares for two reasons,(1) an income effect -
at a lower price Sigma can purchase more shares with the
same budget,and (2) a substitution effect - the increased
expected return at the lower price will make BU shares
more attractive relative to TD shares,Notice that one can
desire a negative number of shares,that is,a short position,
If the stock price is high enough,its expected return will be
so low that the desire to sell will overwhelm diversification
motives and investors will want to take a short position,
Figure 9.2 shows that when the price exceeds $44,Sigma
wants a short position in BU.
9-11
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-12
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
The demand curve for BU shares assumes that
the price and therefore expected return of TD
remain constant,A similar demand curve can be
constructed for TD shares given a price for BU
shares,As before,we would generate the
demand for TD shares by revising Table 9.2 for
various current prices of TD,leaving the price of
BU unchanged,We use the revised expected
returns to calculate the optimal portfolio for each
possible price of TD,ultimately obtaining the
demand curve shown in Figure 9.3.
9-13
资本资产定价模型是现代金融学的奠基石 (风险与期望收益均衡模型 )
It is the equilibrium model that underlies all modern
financial theory.
由诸多简单假定原理来建立
Derived using principles of diversification with
simplified assumptions.
马克维茨,威廉 ·夏普,林特纳和简 ·莫辛研究和发展了资本资产定价模型。
Markowitz,Sharpe,Lintner and Mossin are
researchers credited with its development.
资本资产定价模型
Capital Asset Pricing Model (CAPM)
9-14
１个体投资者是价格的接受者
Individual investors are price takers
２单周期投资期限
Single-period investment horizon
３投资限制在金融资产的交易
Investments are limited to traded financial assets
４无税负和交易成本
No taxes,and transaction costs
假设
Assumptions
9-15
５投资者是理性的均值 -方差完善者
Investors are rational mean-variance optimizers
６同质期望 Homogeneous expectations
给定一系列证券的价格和无风险利率，所有投资者的证券收益的期望收益率与协方差矩阵相等，从而产生了有效率边界和一个独一无二的最优风险资产组合。这一假定也被称为 同质期望。
Given a set of security prices and the risk-free interest rate,
all investors use the same expected returns and covariance
matrix of security returns to generate the efficient frontier and
the unique optimal risky portfolio,This assumption is often
referred to as homogeneous expectations.
对投资者来说信息是无成本的和有效的
Information is costless and available to all investors
假设 Assumptions (cont’d)
9-16
全部投资者将持有相同的风险资产 -市场组合
All investors will hold the same portfolio for risky
assets – market portfolio.
市场组合含有全部股票和每只股票在市场资产组合所占的比例等于它的市值占所有股票的市值
Market portfolio contains all securities and the
proportion of each security is its market value as
a percentage of total market value.
均衡条件
Resulting Equilibrium Conditions
9-17
市场的风险溢价取决于全部市场参与者的平均风险厌恶
Risk premium on the market depends on the
average risk aversion of all market participants
均衡条件
Resulting Equilibrium Conditions (cont.)
式中 σ2 M为市场资产组合的方差； A 为投资者风险厌恶的平均水平。请注意由于市场资产组合是最优资产组合，即风险有效地分散于资产组合中的所有股票,σ2 M也 就是这个市场的系统风险。
9-18
个体证券的风险溢价是市场协方差的函数 Risk
premium on an individual security is a function of
its covariance with the market
贝塔是用来测度股票与一起变动情况下证券收益的变动程度的。贝塔的正式定义如下：
Beta measures the extent to which returns on the
stock and the market move together,Formally,
beta is defined as
均衡条件
Resulting Equilibrium Conditions (cont.)
9-19
个体证券的风险溢价是市场协方差的函数
Risk premium on an individual security is a
function of its covariance with the market
单个证券的风险溢价等于：
The risk premium on individual securities is
均衡条件
Resulting Equilibrium Conditions (cont.)
9-20
当我们把所有个人投资者的资产组合加总起来时，借与贷将互相抵消（这是因为每个借入者都有一个相应的贷出者与之对应），加总的风险资产组合价值等于整个经济中全部财富的价值，这就是市场资产组合。每只股票在这个资产组合中的比例等于股票的市值占所有股票市场价值的比例。
资本资产定价模型认为每个投资者均有优化其资产组合的倾向，最终所有个人的资产组合会趋于一致，每种资产的权重等于它们在市场资产组合中所占的比例。
The portfolios of all individual investors,lending and borrowing will
cancel out (since each lender has a corresponding borrower),and the
value of the aggregate risky portfolio will equal the entire wealth of the
economy,This is the market portfolio,M,The proportion of each stock in
this portfolio equals the market value of the stock (price per share times
number of shares out- standing) divided by the sum of the market values
of all stocks.5 The CAPM implies that as individuals attempt to optimize
their personal portfolios,they each arrive at the same port- folio,with
weights on each asset equal to those of the market portfolio.
市场资产组合
The Market Portfolio
9-21
依据前文给定的假定条件，不难看出所有的投资者均倾向于持有同样的风险资产 组合。如果所有的投资者都将马克维茨分析
(假定 5)应用于同样广泛的证券 (假定 3)，在一个相同的时期内计划他们的投资 (假定 2)，并且投资顺序内容也相同的话 (假定 6),那么他们必然会达到相同的最优风险资产组合。正如下图所示,
Given the assumptions of the previous section,it is easy to
see that all investors will desire to hold identical risky portfolios,
If all investors use identical Markowitz analysis (Assumption 5)
applied to the same universe of securities (Assumption 3) for
the same time horizon (Assumption 2) and use the same input
list (Assumption 6),they all must arrive at the same
determination of the optimal risky portfolio,the portfolio on the
efficient frontier identified by the tangency line from T-bills to
that frontier,as in following figure,
市场资产组合
The Market Portfolio
9-22
资本市场线
Capital Market Line
E(r)
E(rM)
rf
M
资本市场线 CML
m?
9-23
M = 市场组合 Market portfolio
rf = 无风险率 Risk free rate
E(rM) - rf = 市场风险溢价 Market risk premium
E(rM) - rf = 风险市场价格 Market price of risk
= CAPM斜率 Slope of the CAPMM?
市场风险溢价和斜率
Slope and Market Risk Premium
9-24
市场资产组合的均衡风险溢价,E(rM)-rf，与投资者群体的平均风险厌恶程度和市场资产组合的风险 σ 2M
是成比例的。
The equilibrium risk premium on the market
portfolio,E(rM)-rf,will be proportional to
the average degree of risk aversion of the
investor population and the risk of the market
portfolio,Now we can explain this result.
市场资产组合的风险溢价
The Risk Premium of the Market Portfolio
9-25
在简化了的 CAPM模型经济中,无风险投资包括投资者之间的借入与贷出 。 任何借 入头寸必须同时有债权人的贷出头寸作为抵偿 。 这意味着投资者之间的净借入与净贷 出的总和为零 。 那么在风险资产组合上的投资比例总的来说是
100%,或 y = 1。 设 y＝ 1,代入 9-1式经整理,我们发现市场资产组合的风险溢价与风险厌恶的平均水平有关：
In the simplified CAPM economy,risk-free investments involve borrowing
and lending among investors,Any borrowing position must be offset by the
lending position of the creditor,This means that net borrowing and lending
across all investors must be zero,and in consequence the average position
in the risky portfolio is 100%,or y=1,Setting y=1 in equation 9.1 and
rearranging,we find that the risk premium on the market portfolio is related
to its variance by the average degree of risk aversion:
市场资产组合的风险溢价
The Risk Premium of the Market Portfolio
9-26
单个证券的风险益价是单个证券对市场组合风险的贡献函数
The risk premium on individual securities is a
function of the individual security’s contribution
to the risk of the market portfolio.
单个证券的风险益价是构成市场组合资产收益协方差的函数
An individual security’s risk premium is a
function of the covariance of returns with the
assets that make up the market portfolio.
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-27
假定现在我们要测算通用公司股票的资产组合风险，我们用通用公司股票（ GM股） 同市场资产组合的协方差来刻画其对资产组合的风险贡献程度。为解释这种测算方法，
先要再次阐明市场资产组合的方差是如何计算的。为此，
我们按第 8章讨论过的方法将 n阶协方差矩阵各项按照从行到列的顺序分别乘以各证券在市场资产组合中的权重。
Suppose,for example,that we want to gauge the
portfolio risk of GM stock,We mea- sure the contribution
to the risk of the overall portfolio from holding GM stock
by its covariance with the market portfolio,To see why
this is so,let us look again at the way the variance of the
market portfolio is calculated,To calculate the variance
of the market port- folio,we use the bordered covariance
matrix with the market portfolio weights,as discussed in
Chapter 8,We highlight GM in this depiction of the
n stocks in the market portfolio.
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-28
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-29
单个证券的收益和风险
Expected Return and Risk on Individual Securities
通用公司股票对市场资产组合方差的贡献为：
The contribution of GM’s stock to the
variance of the market portfolio is:
9-30
通用公司股票对市场资产组合方差的贡献度市场资产组合的收益率可以表示如下,
The rate of return on the market portfolio may be written as
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-31
通用公司股票与市场资产组合的协方差为,
The covariance of the return on GM with the
market portfolio is,
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-32
测度了通用公司股票对市场方差的贡献度后，我们就可以来确定通用公司股票的合理风险溢价了。首先，我们注意到市场资产组合的风险溢价为 E(rM -rf )，方差为 σ2M,酬报与波动性比率为,
Having measured the contribution of GM stock to market
variance,we may determine the appropriate risk premium for
GM,We note first that the market portfolio has a risk premium
of E(rM -rf ) and a variance of σ2M,for a reward-to-risk ratio of
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-33
假定某位平均的投资者投资于市场资产组合的比例为
100%，现在他打算通过借入无风险贷款的方式来增加比例为小量的市场资产组合头寸。新的资产组合由以下三 部分组成：收益为 rM的原有市场资产组合头寸，
收益为 -δrf 的无风险资产空头头寸，以及收益为 δrM的市场资产组合的多头头寸。总的资产组合收益为 rM＋
δ(rM-rf)，将其期望值与最初期望值 E(rM)比较，期望收益的增加额为单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-34
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-35
单个证券的收益和风险
Expected Return and Risk on Individual Securities
为了度量新资产组合的风险，我们重新计算资产组合的方差。新资产组合由权重为 (1＋ δ)的市场资产组合与权重为 - δ的无风险资产组成，调整后的资产组合的方差为,
To measure the impact of the portfolio shift on risk,
we compute the new value of the portfolio variance,
The new portfolio has a weight of (1+δ) in the market
and -δ in the risk-free asset,Therefore,the variance
of the adjusted portfolio is:
9-36
单个证券的收益和风险
Expected Return and Risk on Individual Securities
由于 δ非常小，所以相比于 2δ而言 δ2可以忽略，
因而我们这里对这一项忽略不计
However,if δ is very small,then δ2 will be
negligible compared to 2δ,so we may ignore
this term
9-37
单个证券的收益和风险
Expected Return and Risk on Individual Securities
综合以上结果，增加的风险溢价与增加的风险之间的平衡，即风险的边际价格为,
Summarizing these results,the trade-off
between the incremental risk premium and
incremental risk,referred to as the marginal
price of risk,is given by the ratio
9-38
单个证券的收益和风险
Expected Return and Risk on Individual Securities
现在，作为一个替代，假定投资者用以无风险利率借入的资金 投资于通用公司股票。他的平均超额收益的增加值为,
Now suppose that,instead,investors were to invest the
increment in GM stock,also financed by
borrowing at the risk-free rate,The increase in mean
excess return is:
9-39
单个证券的收益和风险
Expected Return and Risk on Individual Securities
这一资产组合中投资于市场资产组合的资金权重为 1.0，投资于通用公司股票的资金权重为 δ，
投资于无风险资产的资金权重为 -δ。这一资产组合的方差为,
This portfolio has a weight of 1.0 in the market,
δ in GM,and -δ in the risk-free asset,Its GM
variance is,
9-40
单个证券的收益和风险
Expected Return and Risk on Individual Securities
因此，方差增加值包括两部分：通用公司股票新增头寸的方差和两倍通用公司股票与市场资产组合的协方差：
The increase in variance therefore includes the
variance of the incremental position in GM plus
twice its covariance with the market,
9-41
单个证券的收益和风险
Expected Return and Risk on Individual Securities
δ2忽略不计，通用公司股票的风险边际价格为
Dropping the negligible term involving δ2,the
marginal price of risk of GM is
9-42
单个证券的收益和风险
Expected Return and Risk on Individual Securities
建立通用公司股票的风险边际价格同市场资产组合的风险边际价格相等的等式如下：
Equating the marginal price of risk of GM’s stock to
that of the market results in a relationship between the
risk premium of GM and that of the market:
9-43
单个证券的收益和风险
Expected Return and Risk on Individual Securities
经调整我们得到通用公司股票的正常风险溢价：
To determine the fair risk premium of GM stock,
we rearrange slightly to obtain
9-44
单个证券的收益和风险
Expected Return and Risk on Individual Securities
这里,Cov(rGM,rM)/ σ2M测度的是通用公司股票对市场资产组合方差的贡献程度，这是市场资产组合方差的一个组成部分。这一比率称作贝塔（ beta），以 β表示，这样,9-6式可以写作为：
The ratio Cov(rGM,rM)/ σ2M measures the contribution of GM
stock to the variance of the market portfolio as a fraction of the
total variance of the market portfolio,The ratio is called beta
and is denoted by β,Using this measure,we can restate
equation 9.6 as
9-45
单个证券的收益和风险
Expected Return and Risk on Individual Securities
上式即是 CAPM模型的最普通形式 ─ 期望收益 -贝塔关系,我们对这一关系式还要做更详尽的论述。
This expected return–beta relationship is the most
familiar expression of the CAPM to practitioners,We
will have a lot more to say about the expected return–
beta relationship shortly.
9-46
单个证券的收益和风险
Expected Return and Risk on Individual Securities
If the expected return–beta relationship holds for any individual
asset,it must hold for any combination of assets,Suppose that
some portfolio P has weight wk for stock k,where k takes on
values 1,.,,,n,Writing out the CAPM equation 9.7 for each
stock,and multiplying each equation by the weight of the stock
in the portfolio,we obtain these equations,one for each stock:
9-47
证券市场线
Security Market Line
E(r)
E(rM)
rf
证券市场线
SML
b
bM= 1.0
9-48
b?= [COV(ri,rm)] /?m2
证券市场线斜率 Slope SML = E(rm) - rf
= 市场风险溢价 market risk premium
SML = rf + b[E(rm) - rf]
Betam = [Cov (ri,rm)] /?m2
=?m2 /?m2 = 1
b:证券的协方差风险证券市场线关系
SML Relationships
9-49
β系数。美国经济学家威廉 ·夏普提出的风险衡量指标。用它反映资产组合波动性与市场波动性关系（在一般情况下，将某个具有一定权威性的股指（市场组合）作为测量股票 β值的基准）。
如果 β值为 1.1，即表明该股票波动性要比市场大盘高 10％，说明该股票的风险大于市场整体的风险，当然它的收益也应该大于市场收益，因此是进攻型证券。反之则是防守型股票。无风险证券的 β值等于零，市场组合相对于自身的 β值为 1。
9-50
E(r i ) = rf +b i(E(rm) - rf )
资本资产定价模型的最普通形式 — 期望收益贝塔关系
E(rm) - rf =,08 rf =,03
bx = 1.25
E(rx) =,03 + 1.25(.08) =,13 or 13%
by =,6
E(ry) =,03 +,6(.08) =,078 or 7.8%
证券市场线计算实例
Sample Calculations for SML
9-51
计算图形
Graph of Sample Calculations
E(r)
Rx=13%
SML
b
1.0
Rm=11%
Ry=7.8%
3%
1.25
bx
.6
by
.08
By
9-52
布莱克的零贝塔模型
Black’s Zero Beta Model
缺少无风险资产 Absence of a risk-free asset
在有效边界上的任何资产组合是有效资产组合。
Combinations of portfolios on the efficient frontier are
efficient,
有效率边界上的任一资产组合有不相关组合相伴
All frontier portfolios have companion portfolios that
are uncorrelated,
任何单个资产的收益可以准确地由任意两个边界资产组合的期望收益的线性函数表示。
任何单个资产的收益可以由有效组合的线性函数表示。
Returns on individual assets can be expressed as
linear combinations of efficient portfolios.
9-53
布莱克的零贝塔模型
Black’s Zero Beta Model
布莱克的禁止卖空无风险资产的 CAPM模型建立在下列三项有效率资产组合的方 差均值性质之上：
1) 任何有效率资产组合组成的资产组合仍然是有效率资产组合。
2)有效率边界上的任一资产组合在最小方差边界的下半部分（无效率部分）
上均 有相应的“伴随”资产组合存在，由于这些“伴随”资产组合是不相关的，因此，这 些资产组合可以被视为有效率资产组合中的零贝塔资产组合。
Black’s model of the CAPM in the absence of a risk-free asset rests on the
three following properties of mean-variance efficient portfolios:
1,Any portfolio constructed by combining efficient portfolios is itself on the
efficient frontier.
2,Every portfolio on the efficient frontier has a,companion” portfolio on
the bottom half (the inefficient part) of the minimum-variance frontier with
which it is uncorrelated,Because the portfolios are uncorrelated,the
companion portfolio is referred to as the zero-beta portfolio of the
efficient portfolio.
9-54
有效组合和零贝塔伴随
Efficient Portfolios and Zero Companions
Q
P
Z(Q)
Z(P)
E[rz (Q)]
E[rz (P)]
E(r)
9-55
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
),( ),(),()()()()( 2
QPP
QPPi
QPQi rrC o v
rrC o vrrC o vrErErErE
任何资产的期望收益可以准确地由任意两个边界资产组合的期望收益的线性函 数表示。例如，考虑有两个最小方差边界资产组合 P与 Q，布莱克给出任意资产 i的期 望收益的表达如下：
The expected return of any asset can be expressed as an
exact,linear function of the expected return on any
two frontier portfolios,Consider,for example,the
minimum-variance frontier portfolios P and Q,Black
showed that the expected return on any asset i can be
expressed as
9-56
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
假定经济中只有两个投资者，一个相对来说厌恶风险，而另外一个可以忍受风险。 厌恶风险的投资者选择资本配置线上的资产组合 T，如图 9-5所示，也就是说，他的资 产组合由资产组合 T与按无风险利率贷出的无风险资产组成。 T是由无风险借贷利率 rf 出发的有效率边界的切点。忍受风险的投资者愿意在承担更多风险的前提下取得更高 的风险溢价：他选择图中的 S
。 S资产组合与 T资产组合相比较，虽同处于有效率边界 但其风险与收益均高于 T 资产组合。总的风险资产组合（也就是市场资产组合,M ）由 T与 S结合而成，各自权重由两个投资者的相对财富与风险厌恶程度决定。由于 T与 S 都在有效率边界上
，所以根据性质 1，市场资产组合 M也在有效率边界上。
9-57
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
Imagine an economy with only two investors,one relatively
risk averse and one risk tolerant,The risk-averse investor will
choose a portfolio on the CAL supported by portfolio T in
Figure 9.8,that is,he will mix portfolio T with lending at the
risk-free rate,T is the tangency portfolio on the efficient
frontier from the risk-free lending rate,rf,The risk-tolerant
investor is willing to accept more risk to earn a higher-risk
premium; she will choose portfolio S,This portfolio lies along
the efficient frontier with higher risk and return than portfolio T,
The aggregate risky portfolio (i.e.,the market portfolio,M) will
be a combination of T and S,with weights determined by the
relative wealth and degrees of risk aversion of the two
investors,Since T and S are each on the efficient frontier,so
is M (from Property 1).
9-58
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
9-59
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
根据性质 2，市场资产组合 M 也存在一个在最小方差边界上的零贝塔“伴随”资 产组合,Z(M)，见图 9-5。根据性质 3
及 9-8式，我们可以用市场资产组合 M及 Z(M)来表 示任何证券的收益。由于 Cov(r M,r Z(M) )＝ 0，所以有
From Property 2,M has a companion zero-beta portfolio on
the minimum-variance frontier,Z(M),shown in Figure 9.8,
Moreover,by Property 3 we can express the return on any
security in terms of M and Z(M) as in equation 9.8,
But,since by construction Cov(r M,r Z(M) )＝ 0,the
expression simplifies to
2)()( ),()()()()(
M
Mi
MZMMZi
rrC o vrErErErE
9-60
零贝塔市场模型
Zero Beta Market Model
2)()( ),()()()()(
M
Mi
MZMMZi
rrC o vrErErErE
式中的资产组合 P与资产组合 Q分别由市场资产组合 M及 Z(M)
代替。上式可视为一个简化了的 CAPM模型，在其中,E(r z (m))
取代了 rf 。
where P has been replaced by M and Q has been replaced by
Z(M),Equation 9.9 may be interpreted as a variant of the
simple CAPM,in which r f has been replaced with E(r z (m))
9-61
资本资产定价模型和流动性
CAPM & Liquidity
流动性流动性是指资产转化为现金时所需的费用与便捷程度。交易者非常注重流动性，一些研究证实缺乏流动性将大大降低资产的市场出售价格水平。
Liquidity
Liquidity refers to the cost and ease with which
an asset can be converted into cash,that is,sold,
Traders have long recognized the importance of
liquidity,and some evidence suggests that
illiquidity can reduce market prices substantially,
9-62
资本资产定价模型和流动性
CAPM & Liquidity
非流动溢价 Illiquidity Premium
流动性差的资产低价交易，流动性高的资产期望收益也高，流动性效用的大小同资产的交易费用分布状况以及投资者投资内容的分布有关。
illiquid assets trade at lower prices or,
equivalently,that the expected return on illiquid
assets must be higher,
研究支持非流动溢价
Research supports a premium for illiquidity.
– Amihud and Mendelson
9-63
流动 溢价 的资本资产定价模型
CAPM with a Liquidity Premium
E (ri ) – rf = βi [ E(rM ) - r f ] + f(ci)
f (ci) = 证券 i 的流动溢价
liquidity premium for security i
9-64
非流动性与平均收益关系
Illiquidity and Average Returns
平均月收益率
Average monthly return(%)
买卖差价
Bid-ask spread (%)
9-65
Summary
CAPM 模型假定所有投资者均为单期投资，
并且遵循相同的投资构，并力求获得具有最小方差的最优资产组合。
The CAPM assumes that investors are
single-period planners who agree on a
common input list from security analysis and
seek mean-variance optimal portfolios.
9-66
Summary
CAPM模型假定理想状态下的股票市场具有以下特征：
a,股票市场容量足够大，并且其中所有的投资者为价格接受者。
b,不存在税收与交易费用。
c,所有风险资产均可公开交易。
d,投资者可以以无风险利率借入或贷出任意额度资产。
The CAPM assumes that security markets are ideal in
the sense that:
a,They are large,and investors are price-takers.
b,There are no taxes or transaction costs.
c,All risky assets are publicly traded.
d,Investors can borrow and lend any amount at a fixed
risk-free rate.
9-67
SUMMARY
根据以上假定，投资者持有无差异的风险资产组合。 CAPM模型认为市场资产组合是唯一的具有最小方差的有相切的资产组合，所以消极的投资策略是有效的。
With these assumptions,all investors hold
identical risky portfolios,The CAPM holds
that in equilibrium the market portfolio is the
unique mean-variance efficient tangency
portfolio,Thus a passive strategy is efficient.
9-68
SUMMARY
CAPM 模型中的市场资产组合是市值加权资产组合，其意义为所有股票在资产组合中的权重等于该股票的流通市值占总市值的比重。
The CAPM market portfolio is a value-
weighted portfolio,Each security is held in a
proportion equal to its market value divided
by the total market value of all securities.
9-69
SUMMARY
如果市场资产组合有效且投资者平均无借入或贷出行为，则市场资产组合的风险溢价正比于其方差 σM2，
投资者风险厌恶的平均相关系数 A：
If the market portfolio is efficient and the average
investor neither borrows nor lends,then the risk
premium on the market portfolio is proportional to
its variance,σM2,and to the average coefficient of
risk aversion across investors,A:
E (r M ) – r f = 0.01 A σM2
9-70
SUMMARY
CAPM 模型认为任意单个资产或资产组合的风险溢价为市场资产组合的风险溢 价与贝塔系数的乘积：
The CAPM implies that the risk premium on any individual asset or
portfolio is the product of the risk premium on the market portfolio and
the beta coefficient:
E (r i ) – r f = βi [ E(r M ) - r f ]
这里，贝塔系数等于作为市场资产组合方差一部分的单个资产同市场资产组合的协方差：
where the beta coefficient is the covariance of the
asset with the market portfolio as a fraction of the
variance of the market portfolio
βi=Cov( ri,rM )/σM2
9-71
SUMMARY
在 CAPM模型其他假定不变的条件下，当无风险资产借入或贷出受限制时,CAPM模型的简单形式修正为零贝塔 CAPM模型。零贝塔资产组合期望收益率取代期 望收益 -贝塔关系中的无风险利率：
When risk-free investments are restricted but all
other CAPM assumptions hold,then the simple
version of the CAPM is replaced by its zero-beta
version,Accordingly,the risk-free rate in the
expected return–beta relationship is replaced by
the zero-beta port- folio’s expected rate of return:
E (r i ) = E[ r Z( M) ] + βi E[ r M - r Z( M) ]
9-72
SUMMARY
CAPM 模型的简单形式假定投资者均是短视的行为人。当投资者根据生命期及保留遗产来制定个人投资计划时，只要投资人的偏好及股票收益率分布不变，市场资产组合就仍旧有效，并且 CAPM模型的简单形式及期望收益 -贝塔关系仍然适用。
The simple version of the CAPM assumes that investors
are myopic,When investors are assumed to be concerned
with lifetime consumption and bequest plans,but
investors’ tastes and security return distributions are
stable over time,the market portfolio remains efficient and
the simple version of the expected return–beta
relationship holds.
9-73
SUMMARY
流动费用可以被吸收进 CAPM模型。在存在大量具有贝塔与流动费用 c i 任意组 合的资产的情况下，
期望收益根据下式会哄抬以反映这一非意愿的性质：
Liquidity costs can be incorporated into the
CAPM relationship,When there is a large number
of assets with any combination of beta and
liquidity cost c i,the expected return is bid up to
reflect this undesired property according to
E (r i ) – r f = βi [ E(rM ) - r f ] + f ( c i )
第 9章资本资产定价模型
The Capital Asset Pricing Model
9-2
9.1 股票的需求与均衡价格
9.2 资本资产定价模型
9.3 资本资产定价模型的扩展形式
9.4 资本资产定价模型与流动性资本资产定价模型
Capital Asset Pricing Model (CAPM)
9-3
The supply and demand for shares determine equilibrium
prices and expected rates of return,Imagine a simple world
with only two corporations,Bottom Up Inc,(BU) and Top
Down Inc,(TD),Stock prices and market values are shown
in Table 9.1,Investors can also invest in a money market
fund (MMF) which yields a risk-free interest rate of 5%.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-4
Sigma Fund is a new actively managed mutual fund that has
raised $220 million to invest in the stock market,The security
analysis staff of Sigma believes that neither BU nor TD will
grow in the future and therefore,that each firm will pay level
annual dividends for the foreseeable future,This is a useful
simplifying assumption because,if a stock is expected to pay
a stream of level dividends,the income derived from each
share is a perpetuity,Therefore,the present value of each
share often called the intrinsic value of the share equals the
dividend divided by the appropriate discount rate,A summary
of the report of the security analysts appears in Table 9.2.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-5
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-6
Using these data and assumptions Sigma easily
generates the efficient frontier shown in Figure 9.1
and computes the optimal portfolio proportions
corresponding to the tangency portfolio,These
proportions,combined with the total investment
budget,yield the fund’s buy orders,With a budget of
$220 million,Sigma wants a position in BU of
$220,000,000 X 0.8070 =$177,540,000,or
$177,540,000/39 =4,552,308 shares,and a position
in TD of $220,000,000 X 0.1930= $42,460,000,
which corresponds to 1,088,718 shares.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-7
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-8
The expected rates of return that Sigma used to
derive its demand for shares of BU and TD were
computed from the forecast of year-end stock
prices and the current prices,If,say,a share of
BU could be purchased at a lower price,Sigma’s
forecast of the rate of return on BU would be
higher,Conversely,if BU shares were selling at a
higher price,expected returns would be lower,A
new expected return would result in a different
optimal portfolio and a different demand for shares.
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-9
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-10
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
Sigma’s demand curve for BU stock is given by the
Desired Shares column in Table 9.3 and is plotted in
Figure 9.2,Notice that the demand curve for the stock
slopes downward,When BU’s stock price falls,Sigma will
desire more shares for two reasons,(1) an income effect -
at a lower price Sigma can purchase more shares with the
same budget,and (2) a substitution effect - the increased
expected return at the lower price will make BU shares
more attractive relative to TD shares,Notice that one can
desire a negative number of shares,that is,a short position,
If the stock price is high enough,its expected return will be
so low that the desire to sell will overwhelm diversification
motives and investors will want to take a short position,
Figure 9.2 shows that when the price exceeds $44,Sigma
wants a short position in BU.
9-11
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
9-12
股票的需求与均衡价格
DEMAND FOR STOCKS AND EQUILIBRIUM PRICES
The demand curve for BU shares assumes that
the price and therefore expected return of TD
remain constant,A similar demand curve can be
constructed for TD shares given a price for BU
shares,As before,we would generate the
demand for TD shares by revising Table 9.2 for
various current prices of TD,leaving the price of
BU unchanged,We use the revised expected
returns to calculate the optimal portfolio for each
possible price of TD,ultimately obtaining the
demand curve shown in Figure 9.3.
9-13
资本资产定价模型是现代金融学的奠基石 (风险与期望收益均衡模型 )
It is the equilibrium model that underlies all modern
financial theory.
由诸多简单假定原理来建立
Derived using principles of diversification with
simplified assumptions.
马克维茨,威廉 ·夏普,林特纳和简 ·莫辛研究和发展了资本资产定价模型。
Markowitz,Sharpe,Lintner and Mossin are
researchers credited with its development.
资本资产定价模型
Capital Asset Pricing Model (CAPM)
9-14
１个体投资者是价格的接受者
Individual investors are price takers
２单周期投资期限
Single-period investment horizon
３投资限制在金融资产的交易
Investments are limited to traded financial assets
４无税负和交易成本
No taxes,and transaction costs
假设
Assumptions
9-15
５投资者是理性的均值 -方差完善者
Investors are rational mean-variance optimizers
６同质期望 Homogeneous expectations
给定一系列证券的价格和无风险利率，所有投资者的证券收益的期望收益率与协方差矩阵相等，从而产生了有效率边界和一个独一无二的最优风险资产组合。这一假定也被称为 同质期望。
Given a set of security prices and the risk-free interest rate,
all investors use the same expected returns and covariance
matrix of security returns to generate the efficient frontier and
the unique optimal risky portfolio,This assumption is often
referred to as homogeneous expectations.
对投资者来说信息是无成本的和有效的
Information is costless and available to all investors
假设 Assumptions (cont’d)
9-16
全部投资者将持有相同的风险资产 -市场组合
All investors will hold the same portfolio for risky
assets – market portfolio.
市场组合含有全部股票和每只股票在市场资产组合所占的比例等于它的市值占所有股票的市值
Market portfolio contains all securities and the
proportion of each security is its market value as
a percentage of total market value.
均衡条件
Resulting Equilibrium Conditions
9-17
市场的风险溢价取决于全部市场参与者的平均风险厌恶
Risk premium on the market depends on the
average risk aversion of all market participants
均衡条件
Resulting Equilibrium Conditions (cont.)
式中 σ2 M为市场资产组合的方差； A 为投资者风险厌恶的平均水平。请注意由于市场资产组合是最优资产组合，即风险有效地分散于资产组合中的所有股票,σ2 M也 就是这个市场的系统风险。
9-18
个体证券的风险溢价是市场协方差的函数 Risk
premium on an individual security is a function of
its covariance with the market
贝塔是用来测度股票与一起变动情况下证券收益的变动程度的。贝塔的正式定义如下：
Beta measures the extent to which returns on the
stock and the market move together,Formally,
beta is defined as
均衡条件
Resulting Equilibrium Conditions (cont.)
9-19
个体证券的风险溢价是市场协方差的函数
Risk premium on an individual security is a
function of its covariance with the market
单个证券的风险溢价等于：
The risk premium on individual securities is
均衡条件
Resulting Equilibrium Conditions (cont.)
9-20
当我们把所有个人投资者的资产组合加总起来时，借与贷将互相抵消（这是因为每个借入者都有一个相应的贷出者与之对应），加总的风险资产组合价值等于整个经济中全部财富的价值，这就是市场资产组合。每只股票在这个资产组合中的比例等于股票的市值占所有股票市场价值的比例。
资本资产定价模型认为每个投资者均有优化其资产组合的倾向，最终所有个人的资产组合会趋于一致，每种资产的权重等于它们在市场资产组合中所占的比例。
The portfolios of all individual investors,lending and borrowing will
cancel out (since each lender has a corresponding borrower),and the
value of the aggregate risky portfolio will equal the entire wealth of the
economy,This is the market portfolio,M,The proportion of each stock in
this portfolio equals the market value of the stock (price per share times
number of shares out- standing) divided by the sum of the market values
of all stocks.5 The CAPM implies that as individuals attempt to optimize
their personal portfolios,they each arrive at the same port- folio,with
weights on each asset equal to those of the market portfolio.
市场资产组合
The Market Portfolio
9-21
依据前文给定的假定条件，不难看出所有的投资者均倾向于持有同样的风险资产 组合。如果所有的投资者都将马克维茨分析
(假定 5)应用于同样广泛的证券 (假定 3)，在一个相同的时期内计划他们的投资 (假定 2)，并且投资顺序内容也相同的话 (假定 6),那么他们必然会达到相同的最优风险资产组合。正如下图所示,
Given the assumptions of the previous section,it is easy to
see that all investors will desire to hold identical risky portfolios,
If all investors use identical Markowitz analysis (Assumption 5)
applied to the same universe of securities (Assumption 3) for
the same time horizon (Assumption 2) and use the same input
list (Assumption 6),they all must arrive at the same
determination of the optimal risky portfolio,the portfolio on the
efficient frontier identified by the tangency line from T-bills to
that frontier,as in following figure,
市场资产组合
The Market Portfolio
9-22
资本市场线
Capital Market Line
E(r)
E(rM)
rf
M
资本市场线 CML
m?
9-23
M = 市场组合 Market portfolio
rf = 无风险率 Risk free rate
E(rM) - rf = 市场风险溢价 Market risk premium
E(rM) - rf = 风险市场价格 Market price of risk
= CAPM斜率 Slope of the CAPMM?
市场风险溢价和斜率
Slope and Market Risk Premium
9-24
市场资产组合的均衡风险溢价,E(rM)-rf，与投资者群体的平均风险厌恶程度和市场资产组合的风险 σ 2M
是成比例的。
The equilibrium risk premium on the market
portfolio,E(rM)-rf,will be proportional to
the average degree of risk aversion of the
investor population and the risk of the market
portfolio,Now we can explain this result.
市场资产组合的风险溢价
The Risk Premium of the Market Portfolio
9-25
在简化了的 CAPM模型经济中,无风险投资包括投资者之间的借入与贷出 。 任何借 入头寸必须同时有债权人的贷出头寸作为抵偿 。 这意味着投资者之间的净借入与净贷 出的总和为零 。 那么在风险资产组合上的投资比例总的来说是
100%,或 y = 1。 设 y＝ 1,代入 9-1式经整理,我们发现市场资产组合的风险溢价与风险厌恶的平均水平有关：
In the simplified CAPM economy,risk-free investments involve borrowing
and lending among investors,Any borrowing position must be offset by the
lending position of the creditor,This means that net borrowing and lending
across all investors must be zero,and in consequence the average position
in the risky portfolio is 100%,or y=1,Setting y=1 in equation 9.1 and
rearranging,we find that the risk premium on the market portfolio is related
to its variance by the average degree of risk aversion:
市场资产组合的风险溢价
The Risk Premium of the Market Portfolio
9-26
单个证券的风险益价是单个证券对市场组合风险的贡献函数
The risk premium on individual securities is a
function of the individual security’s contribution
to the risk of the market portfolio.
单个证券的风险益价是构成市场组合资产收益协方差的函数
An individual security’s risk premium is a
function of the covariance of returns with the
assets that make up the market portfolio.
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-27
假定现在我们要测算通用公司股票的资产组合风险，我们用通用公司股票（ GM股） 同市场资产组合的协方差来刻画其对资产组合的风险贡献程度。为解释这种测算方法，
先要再次阐明市场资产组合的方差是如何计算的。为此，
我们按第 8章讨论过的方法将 n阶协方差矩阵各项按照从行到列的顺序分别乘以各证券在市场资产组合中的权重。
Suppose,for example,that we want to gauge the
portfolio risk of GM stock,We mea- sure the contribution
to the risk of the overall portfolio from holding GM stock
by its covariance with the market portfolio,To see why
this is so,let us look again at the way the variance of the
market portfolio is calculated,To calculate the variance
of the market port- folio,we use the bordered covariance
matrix with the market portfolio weights,as discussed in
Chapter 8,We highlight GM in this depiction of the
n stocks in the market portfolio.
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-28
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-29
单个证券的收益和风险
Expected Return and Risk on Individual Securities
通用公司股票对市场资产组合方差的贡献为：
The contribution of GM’s stock to the
variance of the market portfolio is:
9-30
通用公司股票对市场资产组合方差的贡献度市场资产组合的收益率可以表示如下,
The rate of return on the market portfolio may be written as
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-31
通用公司股票与市场资产组合的协方差为,
The covariance of the return on GM with the
market portfolio is,
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-32
测度了通用公司股票对市场方差的贡献度后，我们就可以来确定通用公司股票的合理风险溢价了。首先，我们注意到市场资产组合的风险溢价为 E(rM -rf )，方差为 σ2M,酬报与波动性比率为,
Having measured the contribution of GM stock to market
variance,we may determine the appropriate risk premium for
GM,We note first that the market portfolio has a risk premium
of E(rM -rf ) and a variance of σ2M,for a reward-to-risk ratio of
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-33
假定某位平均的投资者投资于市场资产组合的比例为
100%，现在他打算通过借入无风险贷款的方式来增加比例为小量的市场资产组合头寸。新的资产组合由以下三 部分组成：收益为 rM的原有市场资产组合头寸，
收益为 -δrf 的无风险资产空头头寸，以及收益为 δrM的市场资产组合的多头头寸。总的资产组合收益为 rM＋
δ(rM-rf)，将其期望值与最初期望值 E(rM)比较，期望收益的增加额为单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-34
单个证券的收益和风险
Expected Return and Risk on Individual Securities
9-35
单个证券的收益和风险
Expected Return and Risk on Individual Securities
为了度量新资产组合的风险，我们重新计算资产组合的方差。新资产组合由权重为 (1＋ δ)的市场资产组合与权重为 - δ的无风险资产组成，调整后的资产组合的方差为,
To measure the impact of the portfolio shift on risk,
we compute the new value of the portfolio variance,
The new portfolio has a weight of (1+δ) in the market
and -δ in the risk-free asset,Therefore,the variance
of the adjusted portfolio is:
9-36
单个证券的收益和风险
Expected Return and Risk on Individual Securities
由于 δ非常小，所以相比于 2δ而言 δ2可以忽略，
因而我们这里对这一项忽略不计
However,if δ is very small,then δ2 will be
negligible compared to 2δ,so we may ignore
this term
9-37
单个证券的收益和风险
Expected Return and Risk on Individual Securities
综合以上结果，增加的风险溢价与增加的风险之间的平衡，即风险的边际价格为,
Summarizing these results,the trade-off
between the incremental risk premium and
incremental risk,referred to as the marginal
price of risk,is given by the ratio
9-38
单个证券的收益和风险
Expected Return and Risk on Individual Securities
现在，作为一个替代，假定投资者用以无风险利率借入的资金 投资于通用公司股票。他的平均超额收益的增加值为,
Now suppose that,instead,investors were to invest the
increment in GM stock,also financed by
borrowing at the risk-free rate,The increase in mean
excess return is:
9-39
单个证券的收益和风险
Expected Return and Risk on Individual Securities
这一资产组合中投资于市场资产组合的资金权重为 1.0，投资于通用公司股票的资金权重为 δ，
投资于无风险资产的资金权重为 -δ。这一资产组合的方差为,
This portfolio has a weight of 1.0 in the market,
δ in GM,and -δ in the risk-free asset,Its GM
variance is,
9-40
单个证券的收益和风险
Expected Return and Risk on Individual Securities
因此，方差增加值包括两部分：通用公司股票新增头寸的方差和两倍通用公司股票与市场资产组合的协方差：
The increase in variance therefore includes the
variance of the incremental position in GM plus
twice its covariance with the market,
9-41
单个证券的收益和风险
Expected Return and Risk on Individual Securities
δ2忽略不计，通用公司股票的风险边际价格为
Dropping the negligible term involving δ2,the
marginal price of risk of GM is
9-42
单个证券的收益和风险
Expected Return and Risk on Individual Securities
建立通用公司股票的风险边际价格同市场资产组合的风险边际价格相等的等式如下：
Equating the marginal price of risk of GM’s stock to
that of the market results in a relationship between the
risk premium of GM and that of the market:
9-43
单个证券的收益和风险
Expected Return and Risk on Individual Securities
经调整我们得到通用公司股票的正常风险溢价：
To determine the fair risk premium of GM stock,
we rearrange slightly to obtain
9-44
单个证券的收益和风险
Expected Return and Risk on Individual Securities
这里,Cov(rGM,rM)/ σ2M测度的是通用公司股票对市场资产组合方差的贡献程度，这是市场资产组合方差的一个组成部分。这一比率称作贝塔（ beta），以 β表示，这样,9-6式可以写作为：
The ratio Cov(rGM,rM)/ σ2M measures the contribution of GM
stock to the variance of the market portfolio as a fraction of the
total variance of the market portfolio,The ratio is called beta
and is denoted by β,Using this measure,we can restate
equation 9.6 as
9-45
单个证券的收益和风险
Expected Return and Risk on Individual Securities
上式即是 CAPM模型的最普通形式 ─ 期望收益 -贝塔关系,我们对这一关系式还要做更详尽的论述。
This expected return–beta relationship is the most
familiar expression of the CAPM to practitioners,We
will have a lot more to say about the expected return–
beta relationship shortly.
9-46
单个证券的收益和风险
Expected Return and Risk on Individual Securities
If the expected return–beta relationship holds for any individual
asset,it must hold for any combination of assets,Suppose that
some portfolio P has weight wk for stock k,where k takes on
values 1,.,,,n,Writing out the CAPM equation 9.7 for each
stock,and multiplying each equation by the weight of the stock
in the portfolio,we obtain these equations,one for each stock:
9-47
证券市场线
Security Market Line
E(r)
E(rM)
rf
证券市场线
SML
b
bM= 1.0
9-48
b?= [COV(ri,rm)] /?m2
证券市场线斜率 Slope SML = E(rm) - rf
= 市场风险溢价 market risk premium
SML = rf + b[E(rm) - rf]
Betam = [Cov (ri,rm)] /?m2
=?m2 /?m2 = 1
b:证券的协方差风险证券市场线关系
SML Relationships
9-49
β系数。美国经济学家威廉 ·夏普提出的风险衡量指标。用它反映资产组合波动性与市场波动性关系（在一般情况下，将某个具有一定权威性的股指（市场组合）作为测量股票 β值的基准）。
如果 β值为 1.1，即表明该股票波动性要比市场大盘高 10％，说明该股票的风险大于市场整体的风险，当然它的收益也应该大于市场收益，因此是进攻型证券。反之则是防守型股票。无风险证券的 β值等于零，市场组合相对于自身的 β值为 1。
9-50
E(r i ) = rf +b i(E(rm) - rf )
资本资产定价模型的最普通形式 — 期望收益贝塔关系
E(rm) - rf =,08 rf =,03
bx = 1.25
E(rx) =,03 + 1.25(.08) =,13 or 13%
by =,6
E(ry) =,03 +,6(.08) =,078 or 7.8%
证券市场线计算实例
Sample Calculations for SML
9-51
计算图形
Graph of Sample Calculations
E(r)
Rx=13%
SML
b
1.0
Rm=11%
Ry=7.8%
3%
1.25
bx
.6
by
.08
By
9-52
布莱克的零贝塔模型
Black’s Zero Beta Model
缺少无风险资产 Absence of a risk-free asset
在有效边界上的任何资产组合是有效资产组合。
Combinations of portfolios on the efficient frontier are
efficient,
有效率边界上的任一资产组合有不相关组合相伴
All frontier portfolios have companion portfolios that
are uncorrelated,
任何单个资产的收益可以准确地由任意两个边界资产组合的期望收益的线性函数表示。
任何单个资产的收益可以由有效组合的线性函数表示。
Returns on individual assets can be expressed as
linear combinations of efficient portfolios.
9-53
布莱克的零贝塔模型
Black’s Zero Beta Model
布莱克的禁止卖空无风险资产的 CAPM模型建立在下列三项有效率资产组合的方 差均值性质之上：
1) 任何有效率资产组合组成的资产组合仍然是有效率资产组合。
2)有效率边界上的任一资产组合在最小方差边界的下半部分（无效率部分）
上均 有相应的“伴随”资产组合存在，由于这些“伴随”资产组合是不相关的，因此，这 些资产组合可以被视为有效率资产组合中的零贝塔资产组合。
Black’s model of the CAPM in the absence of a risk-free asset rests on the
three following properties of mean-variance efficient portfolios:
1,Any portfolio constructed by combining efficient portfolios is itself on the
efficient frontier.
2,Every portfolio on the efficient frontier has a,companion” portfolio on
the bottom half (the inefficient part) of the minimum-variance frontier with
which it is uncorrelated,Because the portfolios are uncorrelated,the
companion portfolio is referred to as the zero-beta portfolio of the
efficient portfolio.
9-54
有效组合和零贝塔伴随
Efficient Portfolios and Zero Companions
Q
P
Z(Q)
Z(P)
E[rz (Q)]
E[rz (P)]
E(r)
9-55
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
),( ),(),()()()()( 2
QPP
QPPi
QPQi rrC o v
rrC o vrrC o vrErErErE
任何资产的期望收益可以准确地由任意两个边界资产组合的期望收益的线性函 数表示。例如，考虑有两个最小方差边界资产组合 P与 Q，布莱克给出任意资产 i的期 望收益的表达如下：
The expected return of any asset can be expressed as an
exact,linear function of the expected return on any
two frontier portfolios,Consider,for example,the
minimum-variance frontier portfolios P and Q,Black
showed that the expected return on any asset i can be
expressed as
9-56
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
假定经济中只有两个投资者，一个相对来说厌恶风险，而另外一个可以忍受风险。 厌恶风险的投资者选择资本配置线上的资产组合 T，如图 9-5所示，也就是说，他的资 产组合由资产组合 T与按无风险利率贷出的无风险资产组成。 T是由无风险借贷利率 rf 出发的有效率边界的切点。忍受风险的投资者愿意在承担更多风险的前提下取得更高 的风险溢价：他选择图中的 S
。 S资产组合与 T资产组合相比较，虽同处于有效率边界 但其风险与收益均高于 T 资产组合。总的风险资产组合（也就是市场资产组合,M ）由 T与 S结合而成，各自权重由两个投资者的相对财富与风险厌恶程度决定。由于 T与 S 都在有效率边界上
，所以根据性质 1，市场资产组合 M也在有效率边界上。
9-57
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
Imagine an economy with only two investors,one relatively
risk averse and one risk tolerant,The risk-averse investor will
choose a portfolio on the CAL supported by portfolio T in
Figure 9.8,that is,he will mix portfolio T with lending at the
risk-free rate,T is the tangency portfolio on the efficient
frontier from the risk-free lending rate,rf,The risk-tolerant
investor is willing to accept more risk to earn a higher-risk
premium; she will choose portfolio S,This portfolio lies along
the efficient frontier with higher risk and return than portfolio T,
The aggregate risky portfolio (i.e.,the market portfolio,M) will
be a combination of T and S,with weights determined by the
relative wealth and degrees of risk aversion of the two
investors,Since T and S are each on the efficient frontier,so
is M (from Property 1).
9-58
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
9-59
布莱克的零贝塔模型方程
Black’s Zero Beta Model Formulation
根据性质 2，市场资产组合 M 也存在一个在最小方差边界上的零贝塔“伴随”资 产组合,Z(M)，见图 9-5。根据性质 3
及 9-8式，我们可以用市场资产组合 M及 Z(M)来表 示任何证券的收益。由于 Cov(r M,r Z(M) )＝ 0，所以有
From Property 2,M has a companion zero-beta portfolio on
the minimum-variance frontier,Z(M),shown in Figure 9.8,
Moreover,by Property 3 we can express the return on any
security in terms of M and Z(M) as in equation 9.8,
But,since by construction Cov(r M,r Z(M) )＝ 0,the
expression simplifies to
2)()( ),()()()()(
M
Mi
MZMMZi
rrC o vrErErErE
9-60
零贝塔市场模型
Zero Beta Market Model
2)()( ),()()()()(
M
Mi
MZMMZi
rrC o vrErErErE
式中的资产组合 P与资产组合 Q分别由市场资产组合 M及 Z(M)
代替。上式可视为一个简化了的 CAPM模型，在其中,E(r z (m))
取代了 rf 。
where P has been replaced by M and Q has been replaced by
Z(M),Equation 9.9 may be interpreted as a variant of the
simple CAPM,in which r f has been replaced with E(r z (m))
9-61
资本资产定价模型和流动性
CAPM & Liquidity
流动性流动性是指资产转化为现金时所需的费用与便捷程度。交易者非常注重流动性，一些研究证实缺乏流动性将大大降低资产的市场出售价格水平。
Liquidity
Liquidity refers to the cost and ease with which
an asset can be converted into cash,that is,sold,
Traders have long recognized the importance of
liquidity,and some evidence suggests that
illiquidity can reduce market prices substantially,
9-62
资本资产定价模型和流动性
CAPM & Liquidity
非流动溢价 Illiquidity Premium
流动性差的资产低价交易，流动性高的资产期望收益也高，流动性效用的大小同资产的交易费用分布状况以及投资者投资内容的分布有关。
illiquid assets trade at lower prices or,
equivalently,that the expected return on illiquid
assets must be higher,
研究支持非流动溢价
Research supports a premium for illiquidity.
– Amihud and Mendelson
9-63
流动 溢价 的资本资产定价模型
CAPM with a Liquidity Premium
E (ri ) – rf = βi [ E(rM ) - r f ] + f(ci)
f (ci) = 证券 i 的流动溢价
liquidity premium for security i
9-64
非流动性与平均收益关系
Illiquidity and Average Returns
平均月收益率
Average monthly return(%)
买卖差价
Bid-ask spread (%)
9-65
Summary
CAPM 模型假定所有投资者均为单期投资，
并且遵循相同的投资构，并力求获得具有最小方差的最优资产组合。
The CAPM assumes that investors are
single-period planners who agree on a
common input list from security analysis and
seek mean-variance optimal portfolios.
9-66
Summary
CAPM模型假定理想状态下的股票市场具有以下特征：
a,股票市场容量足够大，并且其中所有的投资者为价格接受者。
b,不存在税收与交易费用。
c,所有风险资产均可公开交易。
d,投资者可以以无风险利率借入或贷出任意额度资产。
The CAPM assumes that security markets are ideal in
the sense that:
a,They are large,and investors are price-takers.
b,There are no taxes or transaction costs.
c,All risky assets are publicly traded.
d,Investors can borrow and lend any amount at a fixed
risk-free rate.
9-67
SUMMARY
根据以上假定，投资者持有无差异的风险资产组合。 CAPM模型认为市场资产组合是唯一的具有最小方差的有相切的资产组合，所以消极的投资策略是有效的。
With these assumptions,all investors hold
identical risky portfolios,The CAPM holds
that in equilibrium the market portfolio is the
unique mean-variance efficient tangency
portfolio,Thus a passive strategy is efficient.
9-68
SUMMARY
CAPM 模型中的市场资产组合是市值加权资产组合，其意义为所有股票在资产组合中的权重等于该股票的流通市值占总市值的比重。
The CAPM market portfolio is a value-
weighted portfolio,Each security is held in a
proportion equal to its market value divided
by the total market value of all securities.
9-69
SUMMARY
如果市场资产组合有效且投资者平均无借入或贷出行为，则市场资产组合的风险溢价正比于其方差 σM2，
投资者风险厌恶的平均相关系数 A：
If the market portfolio is efficient and the average
investor neither borrows nor lends,then the risk
premium on the market portfolio is proportional to
its variance,σM2,and to the average coefficient of
risk aversion across investors,A:
E (r M ) – r f = 0.01 A σM2
9-70
SUMMARY
CAPM 模型认为任意单个资产或资产组合的风险溢价为市场资产组合的风险溢 价与贝塔系数的乘积：
The CAPM implies that the risk premium on any individual asset or
portfolio is the product of the risk premium on the market portfolio and
the beta coefficient:
E (r i ) – r f = βi [ E(r M ) - r f ]
这里，贝塔系数等于作为市场资产组合方差一部分的单个资产同市场资产组合的协方差：
where the beta coefficient is the covariance of the
asset with the market portfolio as a fraction of the
variance of the market portfolio
βi=Cov( ri,rM )/σM2
9-71
SUMMARY
在 CAPM模型其他假定不变的条件下，当无风险资产借入或贷出受限制时,CAPM模型的简单形式修正为零贝塔 CAPM模型。零贝塔资产组合期望收益率取代期 望收益 -贝塔关系中的无风险利率：
When risk-free investments are restricted but all
other CAPM assumptions hold,then the simple
version of the CAPM is replaced by its zero-beta
version,Accordingly,the risk-free rate in the
expected return–beta relationship is replaced by
the zero-beta port- folio’s expected rate of return:
E (r i ) = E[ r Z( M) ] + βi E[ r M - r Z( M) ]
9-72
SUMMARY
CAPM 模型的简单形式假定投资者均是短视的行为人。当投资者根据生命期及保留遗产来制定个人投资计划时，只要投资人的偏好及股票收益率分布不变，市场资产组合就仍旧有效，并且 CAPM模型的简单形式及期望收益 -贝塔关系仍然适用。
The simple version of the CAPM assumes that investors
are myopic,When investors are assumed to be concerned
with lifetime consumption and bequest plans,but
investors’ tastes and security return distributions are
stable over time,the market portfolio remains efficient and
the simple version of the expected return–beta
relationship holds.
9-73
SUMMARY
流动费用可以被吸收进 CAPM模型。在存在大量具有贝塔与流动费用 c i 任意组 合的资产的情况下，
期望收益根据下式会哄抬以反映这一非意愿的性质：
Liquidity costs can be incorporated into the
CAPM relationship,When there is a large number
of assets with any combination of beta and
liquidity cost c i,the expected return is bid up to
reflect this undesired property according to
E (r i ) – r f = βi [ E(rM ) - r f ] + f ( c i )
